Efficiently Answering Probabilistic Threshold Top - k Queries on Uncertain Data ( Extended Abstract )

In this paper, we propose a novel type of probabilistic threshold top-k queries on uncertain data, and give an exact algorithm. More details can be found in [4]. I. PROBABILISTIC THRESHOLD TOP-k QUERIES We consider uncertain data in the possible worlds semantics model [1], [5], [7], which is also adopted by some recent studies on uncertain data processing, such as [8], [2], [6]. Generally, an uncertain table T contains a set of (uncertain) tuples, where each tuple t ∈ T is associated with a membership probability value Pr(t) > 0. When there is no confusion, we also call an uncertain table simply a table. A generation rule on a table T specifies a set of exclusive tuples in the form of R : tr1 ⊕ · · · ⊕ trm where tri ∈ T (1 ≤ i ≤ m) and mi=1 Pr(tri) ≤ 1. The rule R constrains that, among all tuples tr1 , . . . , trm involved in the rule, at most one tuple can appear in a possible world. As [8], [2], we assume that each tuple is involved in at most one generation rule. For a tuple t not involved in any generation rule, we can make up a trivial rule Rt : t. Therefore, conceptually, an uncertain table T comes with a set of generation rules RT such that each tuple is involved in one and only one generation rule in RT . We write t ∈ R if tuple t is involved in rule R. The probability of a rule is the sum of the membership probability values of all tuples involved in the rule, denoted by Pr(R) = ∑ t∈R Pr(t). The length of a rule is the number of tuples involved in the rule, denoted by |R| = |{t|t ∈ R}|. A generation rule R is a singleton rule if |R| = 1. R is a multi-tuple rule if |R| > 1. A tuple is dependent if it is involved in a multi-tuple rule, otherwise, it is independent. For a subset of tuples S ⊆ T and a generation rule R, we denote the tuples involved in R and appearing in S by R ∩ S. A possible world W is a subset of T such that for each generation rule R ∈ RT , |R ∩ W | = 1 if Pr(R) = 1, and |R ∩W | ≤ 1 if Pr(R) < 1. We denote by W the set of all possible worlds. Clearly, for an uncertain table T with a set of generation rules RT , the number of all possible worlds is |W| = ∏ R∈RT ,Pr(R)=1 |R| ∏ R∈RT ,Pr(R)<1(|R| + 1). The number of possible worlds on a large table can be huge. Each possible world is associated with an existence probability Pr(W ) that the possible world happens. Following with the basic probability principles, we have Pr(W ) = ∏ R∈RT ,|R∩W |=1 Pr(R ∩ W ) ∏ R∈RT ,R∩W=∅(1 − Pr(R)). Apparently, for a possible world W , Pr(W ) > 0. Moreover, ∑ W∈W Pr(W ) = 1. A top-k query Q(P, f) contains a predicate P , a ranking function f , and an integer k > 0. When Q is applied on a set of certain tuples, the tuples satisfying predicate P are ranked according to ranking function f , and the top-k tuples are returned. For tuples t1, t2, t1 1f t2 if t1 is ranked higher than or equal to t2 according to ranking function f . 1f , called the ranking order, is a total order on all tuples. Since a possible world W is a set of tuples, a top-k query Q can be applied to W directly. We denote by Q(W ) the top-k tuples returned by a top-k query Q on a possible world W . Q(W ) contains k tuples. A probabilistic threshold top-k query (PT-k query for short) on an uncertain table T consists of a top-k query Q and a probability threshold p (0 < p ≤ 1). For each possible world W , Q is applied and a set of k tuples Q(W ) is returned. For a tuple t ∈ T , the top-k probability of t is the probability that t is in Q(W ) in all W ∈ W , that is, Pr Q,T (t) = ∑ W∈W,t∈Qk(W ) Pr(W ). When Q and T are clear from context, we often write Pr Q,T (t) as Pr (t) for the interest of simplicity. The answer set to a PT-k query is the set of all tuples whose top-k probability values are at least p. That is, Answer(Q, p, T ) = {t|t ∈ T, Pr Q(t) ≥ p}. We are interested in how to compute efficiently the answer set for a PT-k query on an uncertain table. II. AN EXACT ALGORITHM Hereafter, by default we consider a top-k query Q(P, f) on an uncertain table T . P (T ) = {t|t ∈ T ∧ P (t) = true} is the set of tuples satisfying the query predicate. P (T ) is also an uncertain table where each tuple in P (T ) carries the same membership probability as in T . Moreover, a generation rule R in T is projected to P (T ) by removing all tuples from R that are not in P (T ). Then, the problem of answering the PT-k query is to find the tuples in P (T ) whose top-k probability values pass the probability threshold. Apparently, Answer(Q, p, T ) = Answer(Q, p, P (T )). We only need to consider P (T ) in answering a top-k query. A. The Dominant Set Property For a tuple t ∈ P (T ) and a possible world W such that t ∈ W , whether t ∈ Q(W ) depends only on how many other tuples in P (T ) ranked higher than t appear in W . Technically, generation rule R ti compression rule−tuple ti rule−tuple Case 2: ti is ranked lower than all tuples in R rule−tuple, Pr(tR)=Pr(R) Case 3: ti is ranked between tuples in R generation rule R tR_left ti